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					                                              1
Randomized Block Design

In chapter on paired t-tests, we learned to
“match” subjects on variables that:
  influence performance
  but are not of interest.


Matching gives a more sensitive test of H0
because it removes sources of variance that
inflate 2.
                        Lecture 17
                                                     2
Randomized Block Design (RBD)

In the analysis of variance, the matched
subjects design is called the Randomized
Block Design.
  subjects are first put into blocks
     a block is a group matched on some variable
  subjects in a block are then randomly assigned
 to treatments
  for p treatments, you need p subjects per block


                         Lecture 17
                                                  3
Sums of squares in the RBD

We compute SSTreat as before. Compute
SSB (SS for Blocks) analogously:
  Compute deviations of block means from grand
 mean.
  Square deviations, then add them up.


SSTotal is composed of SSTreat + SSE.
Does SSB come from SSTreat or from SSE?
                     Lecture 17
                                         4
Where does SSB come from?


               SST




  SSTotal                       SSB



               SSE
                              Residual
                                SSE
                 Lecture 17
                                                    5
Conceptual Formulas

SST = Σb(XTi – XG)2                    p-1
SSB = Σp(XBi – XG)2                    b-1
SSTotal = Σ(Xi – XG)2                  n-1
SSE = SSTotal – SST – SSB              (b-1)(p-1)
                                       = n-b-p+1
MST = SST/(p-1)
MSB = SSB/(b-1)
MSE = SSE/(b-1)(p-1) = SSE/(n-b-p+1)

                      Lecture 17
                                                             6
Summary table

Source      df     SS                MS          F

Treat        p-1   SSTreat           SST/(p-1)     MST/MSE
Blocks       b-1   SSB               SSB/(b-1)     MSB/MSE
Error    n-p-b+1   SSE               SSE/(n-b-p+1)
Total        n-1   SSTotal




                        Lecture 17
                                              7
Computational Formulas

CM = (ΣX)2             SSTotal = ΣX2 – CM
         n
SSTreat = ΣT2i – CM    SSB = ΣB2i – CM
           b                    p
SSE = SSTotal – SST – SSB

p=# of samples   Ti=Total for ith treatment
b=# of blocks    Bi=Total for ith block


                       Lecture 17
                                                      8
Randomized Block Design – Example 1a

H0: 1 = 2 = 3
HA: At least two differ significantly

Statistical test:        F=           MST
                                      MSE

Rej. region:             Fobt > F(2, 8, .05) = 4.46


                         Lecture 17
                                             9
Randomized Block Design – Example 1a

CM = 104834.4

SSTotal   = ΣX2 – CM
          = 782 + 812 + … + 942 – 104834.4
          = 105198 – 104834.4
          = 363.6


                   Lecture 17
                                       10
Randomized Block Design – Example 1a

SSTreat = Σ(Ti2) – CM
            b
     = 4012 + 4212 + 4322 – 104834.4
         5       5    5
     = 104933.2 – 104834.4
     = 98.8


                  Lecture 17
                                       11
Randomized Block Design – Example 1a

SSB = ΣB2i – CM
       p

SSB = 2442 + … + 2712 – 104834.4
       3          3
    = 105075.33 – 104834.4
    = 240.93

                  Lecture 17
                                       12
Randomized Block Design – Example 1a

SSE = SSTotal – SSTreat – SSB

     = 363.6 – 98.8 – 240.93

     = 23.87




                    Lecture 17
                                                   13
Randomized Block Design – Example 1a

Source     df    SS                MS      F

Treat      2      98.8             49.4    16.55
Blocks     4     240.93            60.23   20.18
Error      8      23.87             2.98
Total      14    363.6

Decision: Reject HO – average scores do differ
across exams.
                      Lecture 17
                                                 14
Randomized Block Design – Example 1b

H0: w = A
HA: W ≠ A

(Note: this is a post-hoc test. We’ll do N-K.)

Statistical test:   Q = Xi – Xj
                       √MSE/n


                         Lecture 17
                                                                   15
 Randomized Block Design – Example 1b

Rank order sample means:

Edison        Wilbur          Orville             Thomas   Alva
 90.3          86              81.3                80.6    79.67

                                            r=4

Qcrit = Q(4, 8, .05) = 4.53


                               Lecture 17
                                                   16
Randomized Block Design – Example 1b

Qobt:

        86 – 79.67   =            6.33    = 6.35
        √(2.984)/3                0.997

Reject HO. Wilbur & Alva differ significantly
on their overall average on the 3 exams.

                     Lecture 17
                                                      17
Randomized Block Design – Example 2a

H0: 1 = 2 = 3
HA: At least two differ significantly

Statistical test:        F=           MST
                                      MSE

Rej. region:             Fobt > F(2, 8, .05) = 4.46


                         Lecture 17
                                              18
Randomized Block Design – Example 2a

CM = 35072      = 819936.6
      15

SSTotal = ΣX2 – CM
        = 2102 + 2452 + … + 2902 – 819936.6
        = 855701 – 819936.6
        = 35764.4


                     Lecture 17
                                           19
Randomized Block Design – Example 2a

SSTreat = Σ(Ti2) – CM
             b
      = 12282 + 11872 + 10922 – 819936.6
          5        5     5
      = 821883.4 – 819936.6
      = 1946.8




                     Lecture 17
                                        20
Randomized Block Design – Example 1a

SSB = ΣB2i – CM
       p

SSB = 6042 + 7272 … + 9582 – 819936.6
       3      3        3
     = 852973.67 – 819936.6
     = 33037.07



                    Lecture 17
                                       21
Randomized Block Design – Example 1a

SSE = SSTotal – SSTreat – SSB

     = 35764.4 – 1946.8 – 33037.07

     = 780.5




                       Lecture 17
                                                     22
Randomized Block Design – Example 1a

Source     df    SS                MS   F

Treat      2     1946.8   973.4         9.977
Blocks     4     33037.07 8259.27       84.656
Error      8     780.5    97.563
Total      14    35764.4

Decision: Reject HO – weights do differ across the
3 time periods.
                      Lecture 17
                                       23
Randomized Block Design – Example 1b




                 Lecture 17

				
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